Nonrotating black hole in a post-Newtonian tidal environment II

TL;DR

This paper constructs a detailed metric for a nonrotating black hole in a tidal environment using post-Newtonian matching, extending previous work to include higher-order tidal moments and gauge choices.

Contribution

It provides a new Regge-Wheeler gauge formulation of the tidally deformed black hole metric and calculates tidal moments at first post-Newtonian order, applicable to various compact objects.

Findings

Derived the black hole metric in Regge-Wheeler gauge up to (r/b)^4 order.

Calculated tidal moments at first post-Newtonian order, independent of the body's nature.

Analyzed the event horizon's geometry under tidal deformation, revealing quadrupolar and octupolar bulges.

Abstract

In the first part of the paper we construct the metric of a tidally deformed, nonrotating black hole. The metric is presented as an expansion in powers of r/b << 1, in which r is the distance to the black hole and b the characteristic length scale of the tidal field --- the typical distance to the remote bodies responsible for the tidal environment. The metric is expanded through order (r/b)^4 and written in terms of a number of tidal multipole moments, the gravitoelectric moments E_{ab}, E_{abc}, E_{abcd} and the gravitomagnetic moments B_{ab}, B_{abc}, B_{abcd}. It differs from the similar construction of Poisson and Vlasov in that the tidal perturbation is presented in Regge-Wheeler gauge instead of the light-cone gauge employed previously. In the second part of the paper we determine the tidal moments by matching the black-hole metric to a post-Newtonian metric that describes a system of bodies with weak mutual gravity. This extends the previous work of Taylor and Poisson (paper I in this sequence), which computed only the leading-order tidal moments, E_{ab} and B_{ab}. The matching is greatly facilitated by the Regge-Wheeler form of the black-hole metric, and this motivates the work carried out in the first part of the paper. The tidal moments are calculated accurately through the first post-Newtonian approximation, and at this order they are independent of the precise nature of the compact body. The moments therefore apply equally well to a rotating black hole, or to a (rotating or nonrotating) neutron star. As an application of this formalism, we examine the intrinsic geometry of a tidally deformed event horizon, and describe it in terms of a deformation function that represents a quadrupolar and octupolar tidal bulge.